ACCELERATED REVIEWS / STUDY COLLECTION
E071
Applied Discrete Mathematics
Logic, counting, modular arithmetic, and graphs
20 WHY APPLY
original questions explained solutions worked examples
Independent course-topic review. Original learning material.
Not affiliated with or endorsed by Western Governors University.
EDITION 1.0 | 29 OCTOBER 2026
,ACCELERATED REVIEWS E071 / INDEPENDENT STUDY GUIDE
Contents
How to use this guide 3
Module 1: Statements, logic, and quantifiers 3
Module 2: Sets, functions, and relations 4
Module 3: Boolean algebra and bit operations 4
Module 4: Counting without double-counting 5
Module 5: Discrete probability 5
Module 6: Modular arithmetic and remainders 6
Module 7: Graphs, trees, and traversal 6
Compact reference 7
Original practice problems 7
Full worked solutions 8
Three-session study plan 9
Glossary 10
References 11
Edition 1.0 - October 2026 2
,ACCELERATED REVIEWS E071 / INDEPENDENT STUDY GUIDE
An independent study guide for E071 topic review. This is original learning material, not an official university
publication or an assessment guide. The public program guide names logic, Boolean algebra, sets, functions,
relations, graphs, combinatorics, and modular arithmetic, with technology applications. The public competency list
also includes probability, which connects naturally to counting. Course materials and assessments can change;
this review does not claim exhaustive current exam alignment or guarantee a passing result.
How to use this guide
Read one module, cover its worked example, and reconstruct it from memory. Then answer its retrieval prompt
aloud. Attempt the 20 practice problems without the solutions; use the worked solutions to locate the exact step
that failed. Write down the corrected rule in your own words. The questions are original, self-contained exercises
designed for learning, not replicas of course assessment items.
Module 1: Statements, logic, and quantifiers
A proposition is a sentence that is either true or false. Let p mean "the request has a valid token" and q mean "the
request is admitted." p AND q requires both conditions; p OR q uses inclusive OR, so either or both suffice. NOT p
reverses the truth value. The conditional p -> q promises only that a true p is followed by a true q. Its sole false
case is p true and q false. A false p makes the conditional true as a logical statement, even when q is also false.
Do not read a conditional as a claim that p is the only route to q.
The converse of p -> q is q -> p; it need not have the same truth value. The contrapositive, NOT q -> NOT p, is
equivalent to the original. This supports a useful inference: from p -> q and NOT q, conclude NOT p (modus
tollens). From p -> q and p, conclude q (modus ponens). A truth table checks a compound statement by listing
each possible assignment of truth values to its distinct propositions. With n independent propositions, there are 2^n
rows. For a two-variable conditional, focus on the four (p, q) cases rather than memorizing a large table.
A predicate becomes a proposition after its variable is specified or quantified. For every x in D, P(x) needs all
members of the stated domain to pass. There exists x in D such that P(x) needs one witness. Negation
switches the quantifier and negates the predicate:
• NOT (for every x, P(x)) means there exists x with NOT P(x).
• NOT (there exists x, P(x)) means for every x, NOT P(x).
Worked example. A policy states, "If a job is approved, it is logged." Suppose an unapproved job is logged. Set p
= approved, q = logged; the observed pair is (false, true). The policy p -> q is true for this pair. It says nothing
about whether all logged jobs were approved. A single approved but unlogged job, (true, false), would violate
the policy.
Common error. Inferring approval from a log entry affirms the consequent: p -> q, q, therefore p. Logging may
have other causes.
Retrieval prompt. What exact truth-value pair falsifies p -> q, and how do you negate "every device is healthy"?
Edition 1.0 - October 2026 3
E071
Applied Discrete Mathematics
Logic, counting, modular arithmetic, and graphs
20 WHY APPLY
original questions explained solutions worked examples
Independent course-topic review. Original learning material.
Not affiliated with or endorsed by Western Governors University.
EDITION 1.0 | 29 OCTOBER 2026
,ACCELERATED REVIEWS E071 / INDEPENDENT STUDY GUIDE
Contents
How to use this guide 3
Module 1: Statements, logic, and quantifiers 3
Module 2: Sets, functions, and relations 4
Module 3: Boolean algebra and bit operations 4
Module 4: Counting without double-counting 5
Module 5: Discrete probability 5
Module 6: Modular arithmetic and remainders 6
Module 7: Graphs, trees, and traversal 6
Compact reference 7
Original practice problems 7
Full worked solutions 8
Three-session study plan 9
Glossary 10
References 11
Edition 1.0 - October 2026 2
,ACCELERATED REVIEWS E071 / INDEPENDENT STUDY GUIDE
An independent study guide for E071 topic review. This is original learning material, not an official university
publication or an assessment guide. The public program guide names logic, Boolean algebra, sets, functions,
relations, graphs, combinatorics, and modular arithmetic, with technology applications. The public competency list
also includes probability, which connects naturally to counting. Course materials and assessments can change;
this review does not claim exhaustive current exam alignment or guarantee a passing result.
How to use this guide
Read one module, cover its worked example, and reconstruct it from memory. Then answer its retrieval prompt
aloud. Attempt the 20 practice problems without the solutions; use the worked solutions to locate the exact step
that failed. Write down the corrected rule in your own words. The questions are original, self-contained exercises
designed for learning, not replicas of course assessment items.
Module 1: Statements, logic, and quantifiers
A proposition is a sentence that is either true or false. Let p mean "the request has a valid token" and q mean "the
request is admitted." p AND q requires both conditions; p OR q uses inclusive OR, so either or both suffice. NOT p
reverses the truth value. The conditional p -> q promises only that a true p is followed by a true q. Its sole false
case is p true and q false. A false p makes the conditional true as a logical statement, even when q is also false.
Do not read a conditional as a claim that p is the only route to q.
The converse of p -> q is q -> p; it need not have the same truth value. The contrapositive, NOT q -> NOT p, is
equivalent to the original. This supports a useful inference: from p -> q and NOT q, conclude NOT p (modus
tollens). From p -> q and p, conclude q (modus ponens). A truth table checks a compound statement by listing
each possible assignment of truth values to its distinct propositions. With n independent propositions, there are 2^n
rows. For a two-variable conditional, focus on the four (p, q) cases rather than memorizing a large table.
A predicate becomes a proposition after its variable is specified or quantified. For every x in D, P(x) needs all
members of the stated domain to pass. There exists x in D such that P(x) needs one witness. Negation
switches the quantifier and negates the predicate:
• NOT (for every x, P(x)) means there exists x with NOT P(x).
• NOT (there exists x, P(x)) means for every x, NOT P(x).
Worked example. A policy states, "If a job is approved, it is logged." Suppose an unapproved job is logged. Set p
= approved, q = logged; the observed pair is (false, true). The policy p -> q is true for this pair. It says nothing
about whether all logged jobs were approved. A single approved but unlogged job, (true, false), would violate
the policy.
Common error. Inferring approval from a log entry affirms the consequent: p -> q, q, therefore p. Logging may
have other causes.
Retrieval prompt. What exact truth-value pair falsifies p -> q, and how do you negate "every device is healthy"?
Edition 1.0 - October 2026 3
